In this paper, we define the table union search problem following previous studies [1]}, [2]}, [3]}.
We start from the data model.
A data lake consists of a collection of tables \(\mathcal {T}\) .
Each table \(T \in \mathcal {T}\)  consists of several columns \(\lbrace t_1, \dots , t_m\rbrace \)  where each column \(t_i\)  can be from different domains.
Here \(m\)  is the number of columns in table \(T\)  (denoted as \(|T| = m\) ).
We will use the notation \(T\)  to denote both the table and its set of columns if there is no ambiguity.
If two columns are semantically close to each other, we regard them as unionable columns.
To determine the unionability between two columns, following previous studies, we employ column encoders to generate the representations of columns.
Then the column unionability score can be computed to measure the relevance between those representations.
A column encoder \(\mathcal {M}\)  takes a column \(t\)  as input and outputs \(\mathcal {M}(t)\)  as the representation.
Given two columns \(t_i\)  and \(t_j\) , the column unionability score is computed as \(\mathcal {F}(\mathcal {M}(t_i), \mathcal {M}(t_j))\) ,
where \(\mathcal {F}\)  is a scoring function between two column representations.
Take a previous study [2]} as an example, the column encoder \(\mathcal {M}\)  for the set, semantic and natural language domain is bag of tokens, bag of tokens plus a mapping from tokens to classes in a knowledge base, and bag of word embedding, respectively;
while the scoring function \(\mathcal {F}\)  is the Jaccard similarity between sets of tokens, that between sets of tokens after transformation and cosine similarity between average word embeddings, respectively.
We will show in Section  how \(\mathsf {Starmie}\)  leverages contrastive learning to obtain an encoder \(\mathcal {M}\) 
that captures richer semantics in columns in the form of contextualized column embeddings .
